Modelling the Volatility-Return Trade-Off When Volatility May Be Nonstationary

CREATES Research Paper 2009-59

63 Pages Posted: 20 Dec 2009

See all articles by Christian M. Dahl

Christian M. Dahl

Department of Business and Economics

Emma M. Iglesias

Michigan State University

Date Written: October 2, 2009


In this paper a new GARCH–M type model, denoted the GARCH-AR, is proposed. In particular, it is shown that it is possible to generate a volatility-return trade-off in a regression model simply by introducing dynamics in the standardized disturbance process. Importantly, the volatility in the GARCH-AR model enters the return function in terms of relative volatility, implying that the risk term can be stationary even if the volatility process is nonstationary. We provide a complete characterization of the stationarity properties of the GARCH-AR process by generalizing the results of Bougerol and Picard (1992b). Furthermore, allowing for nonstationary volatility, the asymptotic properties of the estimated parameters by quasi-maximum likelihood in the GARCH-AR process are established. Finally, we stress the importance of being able to choose correctly between AR-GARCH and GARCH-AR processes: First, it is shown, by a small simulation study, that the estimators for the parameters in an ARGARCH model will be seriously inconsistent if the data generating process actually is a GARCH-AR process. Second, we provide an LM test for neglected GARCH-AR effects and discuss its finite sample size properties. Third, we provide an empirical illustration showing the empirical relevance of the GARCH-AR model based on modelling a wide range of leading US stock return series.

Keywords: Quasi-Maximum Likelihood, GARCH-M Model, Asymptotic Properties, Risk-return Relation

JEL Classification: C12, C13, C22, G12

Suggested Citation

Dahl, Christian M. and Iglesias, Emma M., Modelling the Volatility-Return Trade-Off When Volatility May Be Nonstationary (October 2, 2009). CREATES Research Paper 2009-59. Available at SSRN: or

Christian M. Dahl (Contact Author)

Department of Business and Economics ( email )

Campusvej 55
DK-5230 Odense M
29125486 (Phone)

Emma M. Iglesias

Michigan State University

Agriculture Hall
East Lansing, MI 48824-1122
United States

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